VIX Futures Prices Research Articles

Pricing VIX futures with mixed frequency macroeconomic information

This paper develops a new mixed frequency macro affine stochastic volatility (MacroSV) model and presents a closed-form formula for pricing VIX futures. In our model, the variance is decomposed into long-run and short-run components and the long-run component is driven by the macroeconomic variables. The result from the Monte Carlo experiment shows that the new model outperforms the benchmark model in terms of VIX futures pricing. The results of MacroSV-type models suggest that there is a significant relationship between the volatility and macroeconomic variables. The in-sample forecast results indicate that the models with mixed frequency macroeconomic information outperform the benchmark Heston model. For the out-of-sample pricing, the MacroSV model with the economic information of PPI for the United States has an improvement of 16% over the Heston model.

VIX MODELING FOR A MARKET INSIDER

In this paper, we extend the popular Barndorff–Nielsen–Shephard stochastic volatility model to the case of a pure-jump Ornstein–Uhlenbeck equation with non-vanishing stochastic mean-reversion level. Based on this setup, we derive representations for the squared VIX process and related VIX futures prices. Having these results at hand, we introduce an initially enlarged filtration which models the view of a VIX market insider who has knowledge about the future behavior of the stochastic mean-reversion level of the squared volatility process available. In this enlarged filtration framework, we infer an explicit representation for the anticipative VIX process and obtain the associated time dynamics. We finally investigate the pricing of variance swaps under both backward- and forward-looking information.

Good Volatility, Bad Volatility, and VIX Futures Pricing: Evidence from the Decomposition of VIX

Realized semivariance, computed from intraday positive/negative squared returns, provides an accurate measure of the upside/downside variations of stock returns. This article investigates the role of realized semivariance in pricing the CBOE VIX and VIX futures, using a realized semivariance-based model. We obtain the closed-form pricing formula for the VIX index and VIX futures prices, and show that the new model provides superior pricing performance, both in-sample and out-of-sample. We further analytically derive the pricing formulas for the upside/downside components of the VIX (risk-neutral semivariance). Such a decomposition shows that the information gains from the conventional unsigned realized variance are concentrated on pricing the downside part of the VIX; the new realized semivariance-based model provides a larger and more balanced improvement for both the upside and downside components of the VIX. Our results provide strong evidence that the spread between upside/downside variance is the main driver of the asymmetry in return distributions.

Trader positions in VIX futures

Abstract We investigate the dynamic changes in trader positions of market participants in the VIX futures markets. We find that in a low-VIX period, below the 23.81 threshold determined by our model, changes in VIX futures affect the trading decisions of dealers and leveraged fund managers, but in an opposite direction. During a high-VIX period, dealers and leveraged fund managers would then alter their trading strategies. We highlight the important role of exchange-traded products trading in hedging demand of dealers and show the impact on VIX futures. Trader positions are determinants of VIX futures prices, basis, and VIX premium.

The Law of One Price in Equity Volatility Markets

This paper documents law of one price violations in equity volatility markets. While tightly linked by no-arbitrage restrictions, the prices of VIX futures exhibit significant deviations relative to their option-implied upper bounds. Static arbitrage opportunities occur when the prices of VIX futures violate their bounds. The deviations widen during periods of market stress and predict the returns of VIX futures. A relative value trading strategy based on the deviation measure earns a large Sharpe ratio and economically significant alpha-to-margin. There is evidence that systematic risk and demand pressure contribute to the variation in the no-arbitrage deviations over time.

Do Properly Anticipated Prices Fluctuate Randomly? Evidence from VIX Futures Markets

The VIX index is not traded on the spot market. Hence, in contrast to other futures markets, the VIX futures contract and spot index are not linked by a no-arbitrage condition. The authors examine (1) whether predictability in the VIX index carries over to the futures market and (2) whether there is independent time-series predictability in VIX futures prices. The answer to both questions is no. Samuelson was right: VIX futures prices properly anticipate predictability in volatility and are themselves unpredictable. TOPICS:Futures and forward contracts, volatility measures Key Findings • The authors examine (1) whether predictability in the VIX generates predictability in VIX futures and (2) whether time-series momentum in VIX futures is profitable. • Aside from the obvious importance for understanding price formation, these questions are important for practitioners given the recent spike in VIX, the large and active market in trading volatility and volatility insurance strategies, and the demand for investment advice on what to do. • The short answer to both questions is no, especially after adjusting for risk and realistic transactions costs.

Pricing VIX Futures under the GJR–GARCH Process: An Analytical Approximation Method

This article investigates the performance of GJR-GARCH in pricing VIX futures. We first establish a theoretical relationship between VIX futures price and the model implied VIX, from which an analytical approximation pricing formula is then obtained. We compare the pricing performance of the GJR-GARCH model with the Heston-Nandi model. The results show significant dominance of the GJR-GARCH over the Heston-Nandi in both in-sample and out-of-sample VIX futures pricing. TOPICS:Futures and forward contracts, options Key Findings • A theoretical relationship between VIX futures price and the model implied VIX is established. • An analytical approximation pricing formula for VIX futures under the GJR-GARCH is obtained. • The empirical results show that the analytical approximation pricing method under GJR-GARCH outperforms the analytical pricing method under the Heston-Nandi model.

The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew

We derive the formal expansion of the price of a VIX future (and more generally a VIX power payoff) in various Bergomi models at any order in powers of the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future, which we call implied volatility, and also expand this quantity at any order. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. When the initial term-structure of variance swaps is flat, the expansions are in closed form; otherwise, they involve one-dimensional integrals which are extremely fast to compute. Our extensive numerical experiments show that both the price and implied volatility expansions provide very accurate approximations for a wide range of model parameters which covers the values typically used in the equity and FX derivatives markets and even well beyond---in particular, the expansions are accurate even for very large values of the volatility-of-volatility. They also show that the implied volatility expansion converges much faster than the price expansion. Its truncation at order 1 is already so accurate that it leads to a simple formula for the price of the VIX future that can virtually be considered exact. The derivation of the expansions naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties; it is interesting in itself. The expansions allow us to precisely pinpoint the roles of all the model parameters (volatility-of-volatility, mean reversions, correlations, mixing fraction) in the formation of the prices of VIX futures and VIX power payoffs in Bergomi models. We also use them together with the Bergomi-Guyon expansion of the S&P 500 smile to (instantaneously) calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX2 implied volatility. Our tests and the new expansions shed more light on the inability of traditional stochastic volatility models to jointly fit S&P 500 and VIX market data. The (imperfect but decent) joint fit requires much larger values of volatility-of-volatility and fast mean reversion than the ones previously reported in Bergomi (2005, 2016).

Forecasting and trading on the VIX futures market: A neural network approach based on open to close returns and coincident indicators

Previous work has highlighted the difficulty of obtaining accurate and economically significant predictions of VIX futures prices. We show that both low prediction errors and a significant amount of profitability can be obtained by using a neural network model to predict VIX futures returns. In particular, we focus on open-to-close returns (OTCRs) and consider intraday trading strategies, taking into account non-lagged exogenous variables that closely reflect the information possessed by traders at the time when they decide to invest. The neural network model with only the most recent exogenous variables (namely, the return on the Indian BSESN index) is superior to an unconstrained specification with ten lagged and coincident regressors, which is actually a form of weak efficiency involving markets of different countries. Moreover, the neural network turns out to be more profitable than either a logistic specification or heterogeneous autoregressive models.

An Analytical Formula for Vix Futures and Its Applications: Error Corrections and Replication

Zhu and Lian (Journal of Futures Markets, 2012) proposed the first closed-form formulas for VIX futures prices, whihch are conceptually appealing and easy to implement. Unfortunately, the paper is found to contain three kinds of errors. The main formula (9) for the price of futures misses the factor of 100 in two places. Consequently, the biggest reported RMSE of 3.230 by Zhu-Lian is significantly lower than the correct RMSE of 9.447; the biggest correct in-sample MPE of about 50% implies little practical value of Zhu-Lian. Further, the discretized equation set (14), which is used for estimating parameters, has variables with inconsistent units of time. As a result, the reported parameters could not be replicated. Lastly, the reported counts of VIX futures within three maturity-groups are bafflingly way off by between -26% and 48%. This short note corrects these mistakes, and tries to replicate the empirical work.

Relative Pricing and Risk Premia in Equity Volatility Markets

This paper provides empirical evidence that volatility markets are integrated through the time-varying term structure of variance risk premia. These risk premia predict the returns from selling volatility for different horizons, maturities, and products, including variance swaps, straddles, and VIX futures. In addition, the paper derives a closed-form relationship between the prices of variance swaps and VIX futures. While tightly linked, VIX futures exhibit deviations of varying significance from the no-arbitrage prices and bounds implied by the variance swap market. The paper examines these pricing errors and their relationship to VIX futures’ return predictability.

VIX futures pricing with conditional skewness

We develop a closed‐form VIX futures valuation formula based on the inverse Gaussian GARCH process by Christoffersen et al. that combines conditional skewness, conditional heteroskedasticity, and a leverage effect. The new model outperforms the benchmark in fitting the S&P 500 returns and the VIX futures prices. The fat‐tailed innovation underlying the model substantially reduced pricing errors during the 2008 financial crisis. The in‐ and out‐of‐sample pricing performance indicates that the new model should be a default modeling choice for pricing the medium‐ and long‐term VIX futures.

Modeling VXX

In this paper, we study the VXX Exchange Traded Note (ETN), that has been actively traded in recent years, but has lost 99.84% of its value. Using Zhang's formula for VIX futures prices, we develop the first theoretical model for the VXX that links the SPX, VIX, and VXX. We show that the roll yield of VIX futures drives the difference between the VXX and VIX returns. The roll yield is a mostly negative process. We then provide a simple yet robust estimation of the market price of variance risk using VXX and VIX futures prices. The model can be used to price VXX options.

Do Properly Anticipated Prices Fluctuate Randomly? Evidence from VIX Futures Markets

The VIX index is not traded on the spot market. Hence, in contrast to other futures markets, the VIX futures contract and spot index are not linked by a no-arbitrage condition. We examine (a) whether predictability in the VIX index carries over to the futures market, and (b) whether there is independent time series predictability in VIX futures prices. The answer to both questions is no. Samuelson (1965) was right: VIX futures prices properly anticipate predictability in volatility, and are themselves unpredictable.

VIX Futures Pricing with Conditional Skewness

VIX Futures Pricing with Conditional Skewness

Bounds for VIX futures given S&P 500 smiles

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a family of functionally generated portfolios which often improves the classical bounds while still being tractable; more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

Markets' Notion on Implied Volatility Risks: Insights from Model-Free VIX Futures Pricing

Markets' Notion on Implied Volatility Risks: Insights from Model-Free VIX Futures Pricing

Bounds for VIX Futures Given S&P 500 Smiles

We derive sharp bounds for the prices of VIX futures using the full information of SP more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

A Note on the Premiums and Discounts Embedded in VIX Futures Prices

This article illustrates the volatile nature of the premiums and discounts embedded in the prices of VIX (Chicago Board Options Exchange Market Volatility Index) futures contracts. The fact that the underlying VIX index cannot be traded leads VIX futures to be priced more on expectations of market participants than on a typical cost-of-carry relationship. As they near expiration, VIX futures, in the aggregate, tend to trade at an increased premium, when trading in contango, and at an increased discount, when trading in backwardation. In addition, the premium in these contracts tends to peak as the VIX index nears a low, and the discount in the contracts tends to bottom as the index nears a high.

Volatility Derivatives in Practice: Activity and Impact

We use unique regulatory data to examine open positions and activity in both listed and OTC volatility derivatives. Gross vega notional outstanding for index variance swaps is over USD 2 billion, with dealers short vega in order to supply the long vega demand of asset managers. For maturities less than one year, VIX futures are far more actively traded and have a higher notional amount outstanding than S&P 500 variance swaps. To the extent that dealers take on risk when facilitating trades, we estimate that the long volatility bias of asset managers puts upward pressure on VIX futures prices. Hedge funds have offset this potential impact by actively taking a net short position in nearby contracts. In our 2011‐2014 sample, the net impact added less than half a volatility point, on average, to nearby VIX futures contracts but added between one and two volatility points for contracts in less liquid, longer‐dated parts of the curve. We find no evidence that this price impact forces VIX futures outside no‐arbitrage bounds.